Biophysical and Biomedical Measurement Group, Microsystems and Nanotechnology
Division, Physical Measurement Laboratory, National Institute of Standards
and Technology, Gaithersburg, MD, USA

Our everyday reality is characterized by objective information—information
that is selected and amplified by the environment that interacts with
quantum systems. Many observers can accurately infer that information
indirectly by making measurements on fragments of the environment.
The correlations between the system, \(\mathcal {S}\) , and a fragment, \(\mathcal {F}\) ,
of the environment, \(\mathcal {E}\) , is often quantified by the quantum mutual
information or the Holevo quantity that bounds the classical information
about \(\mathcal {S}\)  transmittable by a quantum channel \(\mathcal {F}\) . The latter is
a quantum mutual information but of a classical-quantum state where
measurement has selected outcomes on \(\mathcal {S}\) . The measurement generically
reflects the influence of the remaining environment, \(\mathcal {E}/\mathcal {F}\) , but
can also reflect hypothetical questions to deduce the structure of
\(\mathcal {S}\mathcal {F}\)  correlations. Recently, Touil et al. examined a different
Holevo quantity, one from a quantum-classical state (a quantum \(\mathcal {S}\) 
to a measured \(\mathcal {F}\) ). As shown here, this quantity upper bounds any
accessible classical information about \(\mathcal {S}\)  in \(\mathcal {F}\)  and can yield
a tighter bound than the typical Holevo quantity. When good decoherence
is present—when the remaining environment, \(\mathcal {E}/\mathcal {F}\) , has
effectively measured the pointer states of \(\mathcal {S}\) —this accessibility
bound is the accessible information. For the specific model of Touil
et al., the accessible information is related to the error probability
for optimal detection and, thus, has the same behavior as the quantum
Chernoff bound. The latter reflects amplification and provides a universal
approach, as well as a single-shot framework, to quantify records
of the missing, classical information about \(\mathcal {S}\) .

The emergence of objective, classical information from quantum systems
is due to amplification: Many pieces of the environment—e.g.,
many photons—each interact with a quantum system and acquire
an imprint of certain states, the pointer states. This is the process
by which select information becomes redundant and accessible to many
different observers. The framework, where the environment decoheres
systems and acts as a communication channel for the resulting information,
is known as quantum Darwinism [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}.
It is the pointer states that survive the interaction with the environment
and create “copies” of themselves from which observers can infer
the pointer state of the system. This process has been seen experimentally
in both natural [21]} and engineered [22]}, [23]}
settings, and both theory and practical calculations are steadily
progressing [24]}, [25]}, [26]}, [27]}, [28]}, [29]}, [30]}, [31]}, [32]}, [33]}, [34]}, [35]}, [36]}, [37]}, [38]}.
